Our novel result for \(\mathcal {C}^{(g)}_{0 \ n.a.}\)  completes the one available in the literature [1]}, [2]}, whose coefficient \(\mathcal {C}^{(g)}_{0,0,2}\)  was determined by considering only the contribution of the term
\(c^{(g)}_{2,0,0,2} \mathcal {I}_{2,0,0,0} \) , coming from the 3-point function, and dropping the other two contributions \(c^{(g)}_{4,0,0,4} \mathcal {I}_{4,0,0,-2} \) , and \(c^{(g)}_{4,0,1,3} \mathcal {I}_{4,0,-1,-1} \) , which are instead coming from the crossed 4-point function.
In particular, our result for the term
proportional to \(\log \left(\frac{-s}{m^2}\right)\) 
in \(\mathcal {C}_{0,0,2}^{(g)}\) , reads:
\(\mathcal {C}_{0,0,2}^{(g)}\bigg |_{\log \left(\frac{-s}{m^2}\right)}&=&\frac{3 i E^2 m^2 }{64 \pi ^2}\log \left(\frac{\bar{\mu }^2}{4E^2}\right)+\frac{E^3 m^3}{16 \pi s}-\frac{i E^2 m^2}{512 \pi ^2}\) 
